For illustration purposes let us now return to
the
\({K}=3\)  scenario
as discussed in [1]}.
The first-step quasi-Hermiticity
(REF ) has
been shown there to imply
the
charge-pseudo-Hermiticity
of the
Hamiltonian
as well as the
parity-pseudo-Hermiticity
of the charge
(see
Table
Nr. 1
in loc. cit.).
The proof was based
on the antilexicographically ordered
relations
(REF )
and on the observation that
\(Z_1\,Z_0=Y_1=Y_1^{\ddagger (1)}\) .
The latter relation is interesting also per se 
because
the operator-product quantity
\(Y_1\)  is of an immediate phenomenological interest in the
random matrix theory [2]} or in the open-system physical
context (cf. Ref. [3]} where such a quantity
has been found conserved).
